Many natural phenomena are characterized by their underlying geometry and topological invariants. Part of understanding such processes is being able to differentiate them and classify them through their topological and geometrical signatures. Many advances have been made which use topological data analysis to such end. In this work we present multiple machine learning tools aided by topological data analysis to classify and understand said phenomena.First, feature extraction from persistence diagrams, as a tool to enrich machine learning techniques, has received increasing attention in recent years. In this paper we explore an adaptive methodology to localize features in persistent diagrams, which are then used in learning tasks. Specifically, we investigate three algorithms, CDER, GMM and HDBSCAN, to obtain adaptive template functions/features. Said features are evaluated in three classification experiments with persistence diagrams. Namely, manifold, human shapes and protein classification. In this area, our main conclusion is that adaptive template systems, as a feature extraction technique, yield competitive and often superior results in the studied examples. Moreover, from the adaptive algorithms here studied, CDER consistently provides the most reliable and robust adaptive featurization.Furthermore, we introduce a framework to construct coordinates in finite Lens spaces for data with nontrivial 1-dimensional $\Z_q := \Z / \Z_q$ persistent cohomology, for $q > 2$ prime. Said coordinates are defined on an open neighborhood of the data, yet constructed with only a small subset of landmarks. We also introduce a dimensionality reduction scheme in $S^{2n−1}/ \Z_q$ (Lens-PCA: LPCA) and demonstrate the efficacy of the pipeline $\Z_q$ -persistent cohomology $\Rightarrow$ $S^{2n−1}/ \Z_q$ coordinates $\Rightarrow$ LPCA, for nonlinear (topological) dimensionality reduction. This methodology allows us to capture and preserve geometrical and topological information through a very efficient dimensionality reduction algorithm.Finally, to make use of some of the most powerful tools in algebraic topology we improve on methodologies that make use of persistent 2-dimensional homology to obtain quasiperiodic scores that indicate the degree of periodicity or quasiperiodicity of a signal. There is a significant computational disadvantage in this approach since it requires the often expensive computation of 2-dimensional persistent homology.Our contribution in this area uses the algebraic structure of the cohomology ring to obtain classes in the 2-dimensional persistent diagram by only using classes in dimension 1, saving valuable computational time in this manner and obtaining more reliable quasiperiodicity scores. We develop an algorithm that allows us to effectively compute the cohomological death and birth of a persistent cup product expression. This allows us to define a quasiperiodic score that reliably separates periodic from quasiperiodic time series.
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